Some interesting counter-examples to things with probability of $0$ occuring

Question

Due to this question, I'm wondering about a list of some interesting examples of when the probability that something was going to occur was $0$ and occurs anyways.

I suppose a really basic example could be that the probability that a random number picked between $1$ and $n$ is prime tends to be $0$ as $n\to\infty$, but there are still an infinite amount of primes.

However, I'm interested in less trivial cases (preferably a list) that might very well blow my mind.

Notice: This is not the same as something impossible to occur nor is it the same as something unlikely to occur. Please see Zero probability and impossibility for some explanation.

Answer 1

There is no way to choose a random integer with a probability distribution uniform on the integers: the probability of any particular integer will be $0$.

Is getting a random integer even possible?

Answer 2

Maybe an interesting example you have not seen: a Wiener process is a continuous version of a random walk. If I generate a Wiener process for $T$ seconds then the probability of generating exactly this process is 0.

Answer 3

If you're looking for an example of a nonempty set of measure zero, that's easy: take the set $\{2,5,8\},$ or the Cantor set.

If you're looking for an example of a "real life" event which has probability zero and happens anyway, forget it: probability zero events don't happen. Here are a couple of fake examples:

"Toss a coin an infinite number of times; whatever sequence of heads and tails comes up is a probability zero event."

Nope. In the real world, there is no such thing as an infinite sequence of coin tosses.

"A continuous random variable has to take some real value, but the probability of any real number is zero."

Nope. In real life, a continuous random variable is never observed to take a real number as its value, it is only observed to land in an interval, which has positive measure.

Answer 4

If the probability of an event a random variable is $0$ and the event is surely in the set of possible outcomes, then the only way this is possible is if there are an infinite amount of possibilities.

As it seems to me you are not interested in any cases where there are an infinite amount of outcomes, then nothing "non-trivial" can be found.

Answer 5

A paradox may provide an example.

The unexpected hanging paradox (described here, for example) describes a process for determining that an event has zero probability of occurring, yet the event still occurs.

This, and similar paradoxes, doesn't require an infinite range of probabilities.

But given the self-contradictory nature of your premise, I doubt that you will discover any example that isn't paradoxical.